Combinatorial and Algebraic Aspects of Varieties

代数组合学的核心理念是，通过确定关键的组合结构，数学的各个方面都可以变得更精确、更具体、更计算可行。这个建议解决了几个具体的问题，这些问题跨越了广泛的数学领域，包括拓扑、代数几何、表示理论、统计学、计算机科学和组合学。中心主题是促进这些领域的计算和理解。这四个研究领域包括组合结构、k-Schur函数、支链聚合物和用于分析有序数据的组合/统计算法。这项工作将在纯数学、理论物理、计算机科学和统计学的几个领域产生广泛的影响。所有提议的工作都将以计算为重点，这将进一步发展算法和证明技术。所有提议的工作都将通过教学和指导本科生、研究生和博士后进行研究，从而对人类产生影响。在教育方面，PI发起了一项服务学习课程，通过数学来解决社区中非营利组织和小企业的问题。

At the heart of algebraic combinatorics is the philosophy that every aspect of mathematics can be made more precise, more concrete and more computationally feasible by identifying key combinatorial structures. This proposal addresses several specific problems that span a wide range of mathematics including topology, algebraic geometry, representation theory, statistics, computer science and combinatorics. The central theme is to facilitate computation and understanding in these areas. The four areas of research include varieties with combinatorial structures, k-Schur functions, branched polymers, and combinatorial/statistical algorithms for analyzing ordered data.The work proposed will have a broad impact in several areas of pure math, theoretical physics, computer science and statistics. All of the proposed work will have a computational focus which will further develop algorithms and proof techniques. All of the proposed work will have a human impact component through teaching and mentoring of undergraduates, graduate students and postdocs doing research. In terms of education, the PI has initiated a service learning course to address problems in non-profit organizations and small businesses in our community through mathematics.